Push Faces Like Sketchup
The Rip Fill tool comes close but it still requires a bit of cleanup to remove hidden faces.
It can be accessed either by pressing space and typing "rip fill", or you can use the keyboard shortcut ALT+v.

In the example you posted, you can now press either x, y, or z to constrain the movement to whatever axis is appropriate.



The squashed and hidden faces may cause you problems so you can remove them by first using Vertex Connect Path ,j, and now Remove Doubles should collapse them.



If you want the nGons, you can select the faces and press f for Make Face.



If the model is at some arbitrary angle,  you can set the Transform Orientation to Normal and then double tap z to move the face along its normal.
If you only tap z once, the movement will still be constrained to the Global axis, the second tap activates the custom Transform Orientation.